Which of the following cannot be valid assignment of probabilities for outcomes of sample space S = {ω₁, ω₂, ω₃, ω₄, ω₅, ω₆, ω₇}
Assignment     ω₁    ω₂    ω₃    ω₄    ω₅    ω₆    ω₇
(a)                   0.1   0.01 0.05  0.03 0.01   0.2    0.6
(b)                   1/7    1/7   1/7    1/7   1/7    1/7   1/7
(c)                   0.1     0.2   0.3    0.4   0.5    0.6   0.7
(d)                  -0.1    0.2    0.3    0.4   -0.2   0.1   0.3
(e)                   1/14  2/14  3/14 4/14  5/14  6/14 15/14

Solution:

(a)

Assignment	ω1	ω2	ω3	ω4	ω5	ω6	ω7
(a)	0.1	0.01	0.05	0.03	0.01	0.2	0.6
(b)	1/7	1/7	1/7	1/7	1/7	1/7	1/7
(c)	0.1	0.2	0.3	0.4	0.5	0.6	0.7
(d)	- 0.1	0.2	0.3	0.4	- 0.2	0.1	0.3
(e)	1/14	2/14	3/14	4/14	5/14	6/14	15/14

Here, each of the numbers p(ω_i) is positive and less than 1.

Sum of probabilities

∑ p (ω_i) = p (ω₁) + p (ω₂) + p (ω₃) + p (ω₄) + p (ω₅) + p (ω₆) + p (ω₇)

= 0.1 + 0.01 + 0.05 + 0.03 + 0.01 + 0.2 + 0.6

= 1

Thus, the assignment is valid.

(b)

ω1	ω2	ω3	ω4	ω5	ω6	ω7
1/7	1/7	1/7	1/7	1/7	1/7	1/7

Here, each of the numbers p(ω_i) is positive and less than 1.

Sum of probabilities

∑ p (ω_i) = p (ω₁) + p (ω₂) + p (ω₃) + p (ω₄) + p (ω₅) + p (ω₆) + p (ω₇)

= 1/7 + 1/7 + 1/7 + 1/7 + 1/7 + 1/7 + 1/7

= 7/7

= 1

Thus, the assignment is valid.

(c)

ω1	ω2	ω3	ω4	ω5	ω6	ω7
0.1	0.2	0.3	0.4	0.5	0.6	0.7

Here, each of the numbers p(ω_i) is positive and less than 1.

Sum of probabilities

∑ p (ω_i) = p (ω₁) + p (ω₂) + p (ω₃) + p (ω₄) + p (ω₅) + p (ω₆) + p (ω₇)

= 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7

= 2.8

≠ 1

Thus, the assignment is not valid.

(d)

ω1	ω2	ω3	ω4	ω5	ω6	ω7
– 0.1	0.2	0.3	0.4	– 0.2	0.1	0.3

Here, p(ω₁) and p(ω₅) are negative, but the probabilities are never negative.

Hence, the assignment is not valid.

(e)

ω1	ω2	ω3	ω4	ω5	ω6	ω7
1/14	2/14	3/14	4/14	5/14	6/14	15/14

Here,

p(ω₇) = 15/14 >1.But any probability lies between 0 and 1 (both inclusive).

Hence, the assignment is not valid

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NCERT Solutions Class 11 Maths Chapter 16 Exercise 16.3 Question 1

Which of the following cannot be valid assignment of probabilities for outcomes of sample space S = {ω₁, ω₂, ω₃, ω₄, ω₅, ω₆, ω₇}
Assignment     ω₁    ω₂    ω₃    ω₄    ω₅    ω₆    ω₇
(a)                   0.1   0.01 0.05  0.03 0.01   0.2    0.6
(b)                   1/7    1/7   1/7    1/7   1/7    1/7   1/7
(c)                   0.1     0.2   0.3    0.4   0.5    0.6   0.7
(d)                  -0.1    0.2    0.3    0.4   -0.2   0.1   0.3
(e)                   1/14  2/14  3/14 4/14  5/14  6/14 15/14

Summary:

The assignments (a) and (b) are valid, whereas the assignments (c), (d), and (e) are NOT valid